Space charge simulation using MAD-X with account of longitudinal motion Valery KAPIN MEPhI & ITEP, Moscow FNAL, Batavia, 31-Mar-2010 MEPhI - Moscow Engineering Physics Institute (State Univ.) ITEP - Institute for Theoretical and Experimental Physics, Moscow

Contents Collaborations: MADX (ITEP & CERN), GSI (SIS18 & SIS100), FNAL (deb. mu2e) Approach & methods overview 4D & 6D benchmarking for SIS18 (GSI, 2009) Applications: DA & Beam losses (TWAC, SIS100) MADX-script algorithm for variable lattice with 6D Debuncher with ORBIT(6x6matr.)-lattice by VN Debuncher with MAD8 (SW)->MADX + BB

ITEP-CERN collaboration on MADX in MAD-X is the successor of MAD-8 (frozen in 2002). MAD-X has a modular organization => Team: custodian (F.Schmidt) + Module Keepers “PTC-TRACK module” is developed by V.Kapin (ITEP) & F.Schmidt (CERN) MAD-X Home Page: “

Initial plans for S.C. in 2006 Two options are discussed I) thin-lens-tracking II) thick-lens (PTC-library): I.a) usage a “linear” kick-matrix for Twiss parameters (b); I.b) a space-charge kicks from "frozen beam" “BB-kicks” during tracking; II) Model using a new PTC elements (many “Beam-Beam”) Presently: BB are in PTC-library (done by E.Forest), but interfacing with MADX -> ??? (manpower required) WE USE ONLY 1-st option for MADX with Sp.Ch. !!!

MADX+ sp.ch. chronology – 4D sp.ch. for DA in TWAC storage ring (ITEP); RuPAC’08 (FRCAU03 at JACOW) 2009 – 4D & 6D sp.ch benchmarking SIS18 (GSI) vs SIMPSONS, MICROMAP codes: report TH5PFP023 at PAC’ – 6D sp.ch beam-loss evaluations for SIS100 (GSI): report TH5PFP023 at PAC’ (Jan-Mar) at FNAL: GSI-MADX-script for variable lattice with 6D for FNAL debuncher (mu2e) > (Apr-July) at GSI: SIS100 tracking with variable multipole contents due at different energies

2. Approaches & methods overview

Thin-lens (option I) “MADX+S.C.” The whole idea is not a new one. For example, references below. Presented methods had been already implemented in other beam dynamics codes. Our task is a step-by-step adaptation some of them to MADX, which is presently one of the most advanced code for nonlinear beam dynamics simulations without space-charge. M. Furman, 1987 PAC, pp ……………………………………………………… Y. Alexahin, 2007 PAC, report code THPAN105.

Features & Algorithm of Direct S.C Simulations with MADX for 4D (coasting beam ) most work is done using macros of MADX input scripts Only 2D(transv.) space-charge fields "Frozen" charge distribution either linear (MATRIX) or Gaussian (BB); Several space-charge kicks within every thick element (bends, quads, drifts etc.);

The 2nd order ray tracing integrator for a number of S-C kicks

Remind about integrators 1) A. Chao, “Adv. Topics”, USPAS, 2000: Thick element with the foc. strength S and the length L: 1 st order “O(L)”: a) drift(L)+kick(SL) ; b) kick(SL)+drift(L) 2 nd order “O(L 2 )”: drift(L/2) + kick(SL) + drift(L/2) 2) MAD-9 project (EPAC’00, TUP6B11): Hamiltonian: H=H ext +H sc => 2 nd order map: M = M ext (t/2) M sc (t) M ext (t/2) Ray Tracing: (L/2n)(SL/n)(L/2n) …. repeat n times

Y.Alexahin, A.Drozhdin, N.Kazarinov, “direct space charge in Booster with MAD8”, Beams-doc-2609-v1 1) direct space charge in MAD using set of BEAMBEAM (BB) elements. Tune shift: where β – the Twiss beta-function at BB location, K – kick acting on the particle, N bb – number of BB elements. 2) The number of particle N in fictitious colliding beam must be set as: Here B f – bunching factor, N – number of particle, C – circumference, L i – distance between successive BB elements,  – relativistic factor. The emittance have been evaluated by fitting the integral of distribution function with: where I – is action variable. 3) Simulation with changing emittance from turn to turn of the beam. Fit at the first turn

Self-consistent (linear) BB-sizes Space-charge kicks simulated by the 1st order MATRIX for linear TWISS calculations; Linearly self-consistent beam sizes calculated by iterations with the TWISS; Analytical Laslett's formula and numerical iterations provide near the same tune shifts (for coasting beam !!!)

BB-sizes for 6D simulations Conte & MacKay, “Intro to Phys. Part. Acc”, 1991, 5.5 Dispersion: trajectories x(s) is composed of two parts: a)betatron oscillations x  (s), and particular solution due to dispersion x D (s)=D(s)  p. b)Statistically beam size is   tot =    + [ D(s)  p ] 2 Beam size in bends ( D x  0)  is increased for beam with  p

Tracking with many BB in 6D S.C. kicks by BB-elements for non-linear tracking; (C.O. shifts are included; a total number BB- elements is not limited); Thin-lens tracking with MADX (similar to MAD8) with lattice conversion by MAKETHIN command Transverse BB-forces are modulated according to longitudinal Gaussian distribution. Two versions: a) given “fake” harmonical oscillations (GSI- 2009) b)  T from real 6D tracking (FNAL-2010)

3. Benchmarking with 4D & 6D simulations using MADX + Sp.Ch. for SIS18 model (GSI, 2009) vs. G. Franchetti (GSI), MICROMAP S. Machida (RAL), SIMPSON G. Franchetti, “Code Benchmarking on Space Charge Induced Trapping”:

Benchmarking SIS18 steps 1-5 for 4D 1) Benchmarking of the Phase Space MADX with BBs

2) Benchmarking of the tunes versus particle amplitude. Sextupole OFF.

3) Benchmarking of the tunes versus particle amplitude, with sextupole ON.

4) Benchmarking of the Tunes versus particle amplitude, with sextupole ON

5) Benchmarking of phase space with space charge and sextupole on at Qx = MAD-X with BB’s MICROMAP SIMPSONS

Benchmarking SIS18 for 6D (fake longitudinal motion) BM-6: Trapping test particle during 1 synchrotron oscillation BM-9: evolution of the transverse rms emittance of a bunch (1000particles)

4. Applications: DA (4D) with Sp.Ch for ITEP’s TWAC storage ring V. V. Kapin, A. Ye. Bolshakov, P. R. Zenkevich, “Influence of space charge on dynamical effects on dynamical aperture of TWAC storage ring”, RuPAC’08 (at JACOW). Dependence of DA on the beam intensity for relative momentum offset DELTAP= Dependence of analytical (Laslett) and computed (MADX-> TWISS) betatron tunes on intensity.

4. Applications: DA of SIS100 (GF-report) “benchmarking” Syst. & rand. errors RMS closed orbit – 1.5/1.0 mm MICROMAP: a) COD=0: central DA=4.05 with 3-  ( ) b) COD=1.5/1.0mm: central 3.3 with 3-  ( ) MADX: a) COD=0: DA=> ( ) b) COD=1.5/1.0mm: DA =>  ( )

4. Applications: Beam loss in SIS100 (GSI) MADX-Lattice with 840 BBs for Sp-charge "BEAM2" for JF (nu_x=-0.090) CDR-2008: Fig.11 Beam loss with space charge for Beam2 (2000 macroparticles). 97% vs 93.7% (?) Initial Mismatching CO + other diff. (seed, sp-ch-center). Initial loss->1.5% => 97% vs 95.2%

5. MADX-script algorithm for 6D-tracking in lattice with variable parameters Normally MADX used as multiturn tracking in lattice with constant parameters: TRACK, recloss, aperture; START, x, px, y, py, t, pt; RUN, turns; ENDTRACK; => Need for multi-runs of one-turn tracking ( turns=1 ), while varying lattice parameters after every turn.

5. MADX-script algorithm (continued) In more details for every turn: a) varying lattice parameters; b) collecting surviving particles and lost particles; c) calculating some “integral” beam parameters; d) reading and filling tables etc. Tool: madx-script language (C-like ) with a quite poor set of commands (no manuals, only a few examples) => a time consuming code developments; CPU memory consuming => run portions with only 25 turns (restarts with a simple external linux-script). The developed script has all attributes of a simple particle tracking code, which can be written and tested with a usual language like FORTRAN in a few weeks. It consists of main code with about 7 hundreds lines, plus macro file with 5 hundreds lines, while the lattice file (MADX-sequence) prepared separately.

CALL Lattice SEQ with BBs CALL Tracking MACROs RESET lattice parameters SET defalult parameters CALL input parameters CALL RUN setup ( ; ) Generate parts distributions at first RUN READ turn-depend. Tables, e.g. K2(n), Qx(n) Create TRACKLOSS table for every RUN Turn_first=1 Create CONSTs table Create table for turn param Read coords; set lost flags yes no READ tables: CONST, RUNINFO READ table for turn param Read coords at last turn Create tables for exchanging data with external EXEc At START TURN loop => SET: variable parameters, e.g. Qx, K2; BB-sizes; BEAM- >NPART; TWISS->Qx, Qy; fill turn-table. TRACK, onepass, recloss, Aperture START commands for all particles RUN, turns=1, maxaper; Save out-coords; count lost parts; set lost flags; fill trackloss table; exclude lost parts Fill tables for external EXEcs and run EXEcs Turn loop

6. Debuncher with ORBIT(6x6matr.)-lattice by V.Nagaslaev Lattice preparation: a) accepting matrices (more 2 hundreds) with specially written FORTRAN code b) generating MAD-X lattice with matrices and special elements (apertures, driving and chr.- correcting sextupoles) c) inserting BB-elements in places between sector matrices => BB-locations (=> type of integrator!) are defined by VN d) Twiss parameters are compared (preserved!).

Simulations with ORBIT by VN Initial DATA: tune shift = ; Ramps are given in tables (Npart in bunch are evaluated as 1.4e12)

Simulations with ORBIT by VN Extraction rate decreased if tune shift is increased

Orbit: X-X’ phase space evolutions at tune shift 0.017)

MADX: Intensity vs turns (  =-0.015) MADX: at turn=3000 Intensity=40%; at turn=9000 Intensity=10% ORBIT: at turn=3000 Intensity=50%; at turn=9000 Intensity=25%

MADX: losses, tunes, sizes vs turns (  =-0.015)

MADX: X-X’ phase space evolutions (  =-0.015)

Remark on coasting to bunched beam transition For coasting 4D beam N=1.75e13 for the tune shift  =-0.015: A good agreement between analitical Laslett’s and numerical TWISS For 6D beam (  z =12m;  p =4.5e-3) due to Dx =>  = Calculated numerically MADX: a) TWISS; b) TBT tracking over 1024 turns => Increase Npart by 2.7 (0.0148/0.054) to reach  = for bunched beam  2 tot =  2  + [ D x (s)  p ] 2 Simple explanation: by evaluating the sums for tune shifts 4D & 6D N part (6D)=N part (4D)* *2.4*(2.5  z / L ring )= 1.75e13*2.4*2.5*12/505= =2.5e12

7. Debuncher with MAD8 (S.Werkema)->MADX + BB Lattice preparation: a) converting MAD8 file to MADX format while preserving all relations for elements excitations created by SW b) inserting BB-elements in MADX lattice according to the 2nd order ray tracing integrator. c) global Twiss parameters are preserved well.

Twiss parameters for 1/3 of ring. MAD8 version 51.15MADX

MADX (mad-lattice): Intensity vs turns (  =-0.015) MADX with MAD-lattice: at turn=3000 Intensity=15% against MADX with ORBIT-matrices: at turn=3000 Intensity=40% Increase Npart by 1.34 (0.0148/0.0110) to reach  = for bunched beam => Npart=1.4e12 (strange -> exact as for ORBIT with matrices)

MADX (mad-lattice): losses, tunes, sizes vs turns (  =-0.015)

MADX (MAD-lattice): X-X’ phase space evolutions (  =-0.015)

Conclusion MADX can be applied for sp-ch simulations for coasting and bunched beams Benchmarking test showed a good agreements with other codes Developed MADX-script for lattice with variable parameters can be used for slow-extraction simulations Discrepancy in results should be studied: a) ORBIT vs MADX using ORBIT-style lattice b) ORBIT-lattice vs MAD-lattice using MADX